In mathematics, a limit is a fundamental concept in calculus and analysis that describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential for defining derivatives, integrals, and continuity.

Basic Definition of a Limit

For a function $f(x)$, the limit as $x$ approaches a value $a$ is written as:

$\lim_{{x \to a}} f(x) = L$

This means that as $x$ gets arbitrarily close to $a$ (from either side), the value of $f(x)$ approaches $L$. Here, $L$ is the limit of $f(x)$ as $x$ approaches $a$.

One-Sided Limits

• Left-hand limit: The limit of $f(x)$ as $x$ approaches $a$ from the left (i.e., values less than $a$) is denoted by:

$\lim_{{x \to a^-}} f(x)$

• Right-hand limit: The limit of $f(x)$ as $x$ approaches $a$ from the right (i.e., values greater than $a$) is denoted by:

$\lim_{{x \to a^+}} f(x)$

A limit exists if and only if both the left-hand and right-hand limits exist and are equal.

Examples

1. Finite Limit:
   Consider $\lim_{{x \to 2}} (3x + 1)$.
   As $x$ approaches 2, the expression $3x + 1$ approaches $3 \cdot 2 + 1 = 7$. Thus,
   $\lim_{{x \to 2}} (3x + 1) = 7.$

2. Infinite Limit:
   Consider $\lim_{{x \to 0}} \frac{1}{x^2}$.
   As $x$ approaches 0, $\frac{1}{x^2}$ grows without bound. Thus,
   $\lim_{{x \to 0}} \frac{1}{x^2} = \infty.$

3. Limit Does Not Exist (DNE):
   Consider $\lim_{{x \to 0}} \frac{1}{x}$.
   As $x$ approaches 0 from the right, $\frac{1}{x}$ tends to $+\infty$, but as $x$ approaches 0 from the left, it tends to $-\infty$. Since the left-hand limit and the right-hand limit are not equal, the limit does not exist.

Tips for Calculating Limits

1. Direct Substitution: Try plugging in the value of $x$ directly. If the function is continuous at that point, this gives the limit.
2. Factorization: If direct substitution results in an indeterminate form like $\frac{0}{0}$, try factoring the numerator and denominator to cancel out terms.
3. Rationalization: For limits involving square roots, multiply by a conjugate to simplify.
4. L'Hopital's Rule: If you encounter indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$, differentiate the numerator and the denominator separately.
5. Squeeze Theorem: If a function is "squeezed" between two other functions that have the same limit at a point, then it has the same limit at that point.

Would you like more details on a specific type of limit or a related concept?

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Related Questions:

1. What are some common indeterminate forms in limits?
2. How do you compute the limit of a sequence?
3. What is the formal epsilon-delta definition of a limit?
4. How does continuity relate to limits?
5. How do you find limits involving trigonometric functions?

Tip: To master limits, practice solving a variety of problems using different methods, such as factorization, L'Hopital's rule, and the squeeze theorem.